The overall long-term goal of this proposed research is to develop efficient algorithms which permit comprehensive computer simulation of renal function. It is proposed to continue the research that during the last nine years has led to very useful mathematical models of the medullary counter-flow system and the whole kidney. To make our models more comprehensive and realistic (e.g., simulate tracer washout curves and estimate membrane parameters), further improvements are required in our computer algorithms for the solution of the complicated systems of coupled, stiff, and nonlinear differential equations describing the models. It is hoped to increase the accuracy and stability of the algorithms through the use of piecewise polynomial approximations, Implicit Runge-Kutta methods, and optimum basis functions obtained from the analytic local solutions of conceptually simplified models, when transforming the model differential equations to their finite difference (or finite element) analogues. The use of parallel and multiple shooting, collocation, Rayleigh-Ritz and Galerkin, Invariant-Imbedding, Chebyshev expansion methods will be investigated. It is intended to achieve significant improvements in the efficiency of the algorithms by partitioning the equations in accordance with the physiological connectivity of the kidney, tearing the equations and variables that correspond to the solutes with minor effects on the water flow, the application of sparse matrix techniques, and the use of quasi Newton and projection methods to minimize the computer storage and run-times. It is expected that as a result of the continued successful application of the above techniques the models will progressively improve in sophistication and permit a serious attack on the problem of parameter estimation. For this purpose the use of experimental values, continuation, smoothing, minimization of energy requirements and related techniques is proposed. The methods already developed, as well as currently available experimental information about the parameters, flows, pressures, concentrations and architecture will be progressively incorporated in to our multinephron, multisolute, path following models.